We considered a convex bilevel optimization problem when the outer level problem was to find a minimizer of a strongly convex function over the set of solutions of the inner level problem which was in the form of minimization of the sum of a convex differentiable function and a nonsmooth convex function. In this work, we proposed a novel accelerated algorithm by employing both linesearch and inertial techniques for solving a convex bilevel optimization problem. Then, we proved the strong convergence of the sequence generated by our proposed algorithm to an optimal solution of the convex bilevel optimization problems without the continuity assumption on the gradient of the objective function. Moreover, we presented the convergence behavior of the proposed method by some numerical experiments addressing image restoration problems and data classification problems with least squares constraints. Finally, the performances of the restorative image and the data classification of the proposed method were compared with other existing algorithms in the literature. According to the experiment, our proposed algorithm had a better convergence behavior than the others in the literature.

In this paper, we propose a new accelerated algorithm for solving convex bilevel optimization problems using some fixed point and two-step inertial techniques. Our focus is on analyzing the convergence behavior of the proposed algorithm. We establish a strong convergence theorem for our algorithm under some control conditions. To demonstrate the effectiveness of our algorithm, we utilize it as a machine learning algorithm to solve data classification problems of some noncommunicable diseases, and compare its efficacy with BiG-SAM and iBiG-SAM.

We formulate and analyze a within-host HIV-CTL model with two biologically motivated discrete delays: an intracellular eclipse delay associated with viral production and a delay in cytotoxic T-lymphocyte activation. While each delay has been studied separately, their combined influence on threshold structure and delay-dependent stability has not been systematically examined in a unified framework. We establish positivity and boundedness of solutions, characterize the infection-free and endemic equilibria, and derive the basic reproduction number together with delay-dependent stability conditions and a Hopf-type transition criterion for the endemic state. Numerically, we perform coarse and high-resolution delay sweeps and construct a two-parameter stability map using viral-load oscillation amplitude as the main diagnostic output. Under the biologically calibrated baseline parameter set, solutions converge to a non-oscillatory chronic state across the physiologically relevant delay ranges explored, and the computed oscillation amplitudes remain numerically negligible. Sensitivity screening and robustness checks further indicate that the infection rate and the CTL activation rate are the dominant drivers of long-term viral burden. These results clarify how intracellular and immune-activation delays shape HIV-CTL dynamics, while showing that delay-induced instability, although possible in principle, is not numerically detected in the baseline regime considered here.